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I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a ...

I am studying the compactness of some convolution operators. Let the convolution $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a $C_0$-semigroup on some ...

I am dealing with the following equation: $$ f(x) = g(x) + \intop_{X} dt K(x-t)f(t) \;,\qquad \left\lbrace \begin{array}{c}f(x)>0\;,\;x\in X \\ f(x)<0\;,\;x\notin X \end{array}\right.$$ where $X$...